Weakly prime ideals issued from an amalgamated algebra

Abstract

Let $R$ be a commutative ring with identity. A proper ideal $P$ is said to be weakly prime ideal of $R$ if for every $0\\neq ab\\in P$ where $a,b\\in R,$ implies $a\\in P$ or $b\\in P$. The notion of weakly prime ideal was introduced by Anderson et al. in [Weakly prime ideals,\xa0 Houston J. Math., 2003] as a generalization of prime ideals. The purpose of this paper is to study the form of weakly prime ideals of amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A\\bowtie^{f}J$), introduced and studied by D Anna et al. in [Amalgamated algebras along an ideal, Commutative Algebra and Its Applications, 2009]. Our results provide new techniques for the construction of new original examples of weakly prime ideals. Furthermore, as an application of our results, we provide an upper bound for the weakly Krull dimension of amalgamation.